A semilinear heat equation with concave-convex nonlinearity

Abstract

In this paper, we are interested in the parabolic equation u_t − ∆u = λu^q + u^p in a bounded domain of IR^N, with the Dirichlet boundary condition and the parameters 0 0. We study the initial value problem and the global behavior of the the positive solutions. We are mainly interested in the relations between the global (in time) solutions of the parabolic equation and the solutions of the stationary, elliptic problem. We show in particular that there exists a global solution if and only if there exists a weak solution of the stationary equation.

Keywords

• nonlinear
• parabolic
• elliptic
• initial value problem
• global solution
• finite-time blow up
• weak solution
• supersolution